论文标题
拉丁语tableaux的同位素图
Isotopy graphs of Latin tableaux
论文作者
论文摘要
拉丁语tableaux是拉丁广场的概括，该广场首次出现在2000年代初期的《 Chow》，《粉丝》，《戈曼人》和《Vondrák》中。在这里，我们将同位素的概念（一种排列组的动作）从拉丁正方形到拉丁语tableaux扩展。我们定义了拉丁语tableaux的同位素图，该图在同位素作用下编码轨道的结构，并研究了拉丁图表的形状与其同位素图的结构之间的关系。我们的主要结果表明，对于任何积极的整数$ d $，都有一个拉丁图tableau的同位素图是$ d $维的立方体。我们表明，大多数同位素图不含三角形，并且给出了所有拉丁tableaux的表征，同位素图包含一个三角形。我们还为同位素图的每个组件的顶点的程度提供了一个公式，该公式既取决于拉丁语tableaux和填充的形状。
Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondrák. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relationship between the shape of a Latin tableau and the structure of its isotopy graph. Our main result shows that for any positive integer $d$, there is a Latin tableau whose isotopy graph is a $d$-dimensional cube. We show that most isotopy graphs are triangle-free, and we give a characterization of all the Latin tableaux for which the isotopy graph contains a triangle. We also give a formula for the degree of a vertex of each component of an isotopy graph which depends on both the shape of the Latin Tableaux and the filling.
